Problem 49 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ -4(y+3)>0 $$

Step-by-Step Solution

Answer

Question: Solve the inequality and provide the solution: $$-4(y+3) > 0$$ Answer: $$y < -3$$

Step 1: Distribute the -4 on the left side of the inequality.

We have the inequality, $$-4(y+3) > 0$$. Multiply the left side by the -4: $$-4y - 12 > 0$$.

Step 2: Isolate y variable.

Add 12 to both sides of the inequality to isolate y on the left side. $$-4y - 12 + 12 > 0 + 12$$ $$-4y > 12$$

Step 3: Divide by -4.

Divide both sides by -4. Remember to flip the inequality sign because we are dividing by a negative number. $$\frac{-4y}{-4} < \frac{12}{-4}$$ $$y < -3$$

Step 4: Final Answer

The solution to the inequality is: $$y < -3$$

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows you to simplify expressions by distributing a multiplier across terms inside a parenthesis. In the given equation, we have

• \(-4(y+3)\).

Here, we use the distributive property to multiply

• \(-4\) by \(y\) and \(3\) separately.

This results in:

• \(-4y - 12\).

By applying this property, we can remove parentheses and make complex expressions simpler. With the distributive property, you're essentially breaking down a larger problem into smaller, manageable chunks that are easier to handle.

Isolation of Variables

When solving inequalities or equations, one of the primary goals is to isolate the variable you're solving for, which in this case is

• \(y\).

By isolating \(y\), you can clearly understand what set of values \(y\) can take. In the step-by-step solution, after distributing, we have

• \(-4y - 12 > 0\).

To isolate \(y\), add \(12\) to each side of the inequality:

• \(-4y - 12 + 12 > 0 + 12\).

This simplifies to

• \(-4y > 12\).

Now, \(y\) is one step away from being alone on one side of the inequality, which is crucial for determining its values. Isolation of variables makes mathematical equations easier to work with and is a foundational skill in solving inequalities.

Flipping Inequality Signs

Inequalities and equations are similar but there's one critical difference. When working with inequalities, if you multiply or divide each side by a negative number, you need to flip the inequality sign. This is a key principle to keep in mind.

The reasoning lies in the number line; multiplying or dividing by a negative reverses the sense of the inequality because it essentially "flip" every number's position relative to zero.

In our solution, we have

• \(-4y > 12\).

To solve for \(y\), we divide both sides by

• \(-4\),

resulting in:

• \(y < -3\).

Notice how the ">" changes to "<". Understanding when and why the inequality sign changes is important for accuracy in solving these types of problems.

Negative Numbers in Inequalities

Negative numbers take a prominent role in inequalities, especially when you are tasked with multiplying or dividing. They directly affect the inequality's direction. The example,

• \(-4(y+3) > 0\),

demonstrates the impact of negative numbers.

Here, \(-4\) is multiplied across the parentheses, maintaining the same inequality direction. However, when you reach the step

• \(-4y > 12\),

dividing both sides by a negative affects the direction of the inequality.

• \(y < -3\)

is now the solution instead of \(y > -3\).

It's important to remember that negative numbers "flip" the meaning between greater than and less than. This can make a significant difference in the solution, which is why careful attention to sign changes is necessary when solving inequalities.